How can I rotate the pivot point of my object without rotating the object itself?
After some headscratching about what you mean, I figured it out. Use the shortcut CTRL+A, then apply "Rotation and Scale"



EDIT
What this does is it simply applies your rotation and scale of the object, aligning the local Z axis with the global Z axis and so on. It also applies the scale of the object, meaning textures will align properly.

Hope this helps.